Almost reducibility and oscillatory growth of Sobolev norms
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866913194640932864 |
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| author | Liang, Zhenguo Zhao, Zhiyan Zhou, Qi |
| author_facet | Liang, Zhenguo Zhao, Zhiyan Zhou, Qi |
| contents | For 1D quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form of $(x,-{\rm i}\partial_x)$, we show its almost reducibility. The growth of Sobolev norms of solution is described based on the scheme of almost reducibility. In particular, an $o(t^s)-$upper bound is shown for the $\CH^s-$norm if the equation is non-reducible. Moreover, by Anosov-Katok construction, we also show the optimality of this upper bound, i.e., the existence of quasi-periodic quadratic perturbation for which the growth of ${\mathcal H}^s-$norm of the solution is $o(t^s)$ as $t\to\infty$ but arbitrarily ``close" to $t^s$ in an oscillatory way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_06814 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Almost reducibility and oscillatory growth of Sobolev norms Liang, Zhenguo Zhao, Zhiyan Zhou, Qi Analysis of PDEs Dynamical Systems For 1D quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form of $(x,-{\rm i}\partial_x)$, we show its almost reducibility. The growth of Sobolev norms of solution is described based on the scheme of almost reducibility. In particular, an $o(t^s)-$upper bound is shown for the $\CH^s-$norm if the equation is non-reducible. Moreover, by Anosov-Katok construction, we also show the optimality of this upper bound, i.e., the existence of quasi-periodic quadratic perturbation for which the growth of ${\mathcal H}^s-$norm of the solution is $o(t^s)$ as $t\to\infty$ but arbitrarily ``close" to $t^s$ in an oscillatory way. |
| title | Almost reducibility and oscillatory growth of Sobolev norms |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2208.06814 |